We consider interactions between the two most important components of the atmosphere and ocean dynamics: slowly evolving vortical motion and inertia-gravity waves in rotating stratified axisymmetric flows. Any steady axisymmetric solution for a finite volume anticyclonic vortex with outcropping isopycnals is known to correspond to a set of self-similar analytical time-periodic pulson solutions assuming flows in surrounding fluid is negligible. Here we analyze the flow patterns generated in homogeneous fluid below stratified pulsating lens-like vortex and its feedback on the upper layer vortex

SUTYRIN, Georgi

ZEITLIN, Vladimir

I-Revues

18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007 
1 
Generation of Inertia-Gravity Waves by  
Pulsating Lens-like Axisymmetric Vortices 
 
Georgi Sutyrin & Vladimir Zeitlin 
 
Graduate School of Oceanography  
University of Rhode Island,  
Narragansett, RI 02882 USA 
gsutyrin@gso.uri.edu 
 
Laboratoire de Meteorologie Dynamique  
Ecole Normale Superieure  
24 Rue Lhomond 75231 Paris FRANCE 
zetlin@lmd.ems.fr 
 
Abstract  : 
 
We consider interactions between the two most important components of the atmosphere and ocean 
dynamics: slowly evolving vortical motion and inertia-gravity waves in  rotating stratified axisymmetric 
flows. Any steady axisymmetric solution for a finite volume anticyclonic vortex with outcropping 
isopycnals is known to correspond to a set of self-similar analytical time-periodic pulson solutions 
assuming flows in surrounding fluid is negligible. Here we analyze the flow patterns generated in 
homogeneous fluid below stratified pulsating lens-like vortex and its feedback on the upper layer vortex. 
 
Résumé : 
 
Nous considérons les interactions entre les deux composantes de la dynamique océanique et/ou 
atmosphérique les plus importantes: les mouvements tourbillonnaires lentes et les ondes d'inertia-gravité 
rapides dans les écoulements stratifiés en rotation. Il est connu que chaque solution stationnaire 
axisymétrique pour un vortex anti-cyclonique aux surface isopycnes en intersection avec la surface libre 
engendre une série de solutions auto-similaires, périodiques en temps (“pulsons”), si le mouvement du 
reste du fluide, supposé homogène, est négligée. Nous analysons ci-dessous les écoulements générées 
dans le fluide profond homogène au-dessous d'un pulson, et sa réaction inverse sur le vortex dans les 
couches supérieures. 
 
Key-words :  baroclinic vortex ; rotating fluid ; inertia-gravity waves 
 
1 Introduction 
 
        Hydrostatic, stratified Boussinesque primitive equations (PE) are widely used for modeling 
large- and meso-scale variability in planetary atmospheres and oceans. Solutions for the PE 
include the two most important components of the atmosphere and ocean dynamics: slowly 
evolving vortical motion and inertia-gravity waves (IGW). At the synoptic scale and larger, the 
later are relatively weak, giving rise to a nearly "balanced" vortical dynamics, usually related to 
the potential vorticity inversion (e.g., see Sutyrin 2004 and references therein). The problem of 
emitting IGW and adjustment to balanced state has been considered in a number of publications.  
In particular, the nonlinear geostrophic adjustment of single-scale vortex-like disturbances in 
rotating shallow water was analyzed in detail by Reznik et al. (2001) and in continuously 
stratified fluid by Zeitlin et al. (2003) by means of the multi-time-scale perturbation theory in 
the Rossby number. Although the classical scenario of adjustment was, generally, confirmed, it 
was also demonstrated that large-scale large-amplitude initial perturbations contain near-inertial 
oscillations which stay coupled to the slow vortical component of the flow for a long time.   
 
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007 
2 
        Exact analytic nonlinear solutions for finite-area lens-like vortices pulsating with inertial 
frequency (pulsons) were recently described in a self-similar form (see Sutyrin 2006 and 
references therein). Here we analyze the effect of deep lower layer on the evolution of the 
upper-layer pulson using asymptotic expansions. The rest of the paper is organized as follows. 
In section 2 we formulate primitive equations for axisymmetric flows in the stratified upper 
layer overlying the homogeneous lower layer. In section 3 we discuss a special case of inertially 
pulsating vortices when the solution remains self-similar with the lower layer at rest. In section 
4 we analyze the case with active lower layer. Section 5 provides a summary and conclusions. 
 
2 Model Formulation 
 
We consider a stratified, Boussinesq fluid on the rotating plane. Assuming axisymmetry, we 
write the governing equations for an inviscid flow with the velocity (U, V, W) in the cylindrical 
coordinates (r, θ, z), 
 
4
)( 2
3
22
rf
r
MfV
r
V
r
pP
z
UW
r
UU
t
U
−=+=∂
+∂
+∂
∂
+∂
∂
+∂
∂
,                                       (1) 
 
0)( =∂
∂
+∂
∂
+∂
∂
=++∂
∂
+∂
∂
+∂
∂
z
MW
r
MU
t
MUf
r
V
z
VW
r
VU
t
V
,                                        (2) 
 
0=∂
∂
+∂
∂
+
z
W
r
U
r
U
,                                                                                                                         (3) 
 
0=
∂
∂⎟⎠
⎞⎜⎝
⎛
∂
∂
+
∂
∂
+
∂
∂
z
P
z
W
r
U
t
 ,                                                                                                        (4) 
 
where f is the Coriois parameter, M = Vr + fr²/2 is the absolute angular momentum which is 
conserved by fluid parcels as well as the buoyancy, g 00 /)( ρρρ −  = zP ∂∂ /  related to the 
geopotential, P + p, by the hydrostatic approximation, g is the gravity acceleration, ρ is the 
density, 0ρ is its reference value, and p is the geopotential in the deep homogeneous layer 
where the horizontal velocity (u, v) does not depend on the depth: 
 
fv
r
v
r
p
r
u
u
t
u
+=∂
∂
+∂
∂
+∂
∂ 2
,                  0)( =++∂
∂
+∂
∂
uf
r
v
r
v
u
t
v
.                                           (5) 
 
In the rigid-lid approximation the deep radial velocity can be expressed by the velocity in 
the upper stratified layer from the total mass balance by integrating (3) 
 
∫−=− hUdzuhD 0)(  ,                                                                                                      (6) 
 
where h(r, t) is the depth of the upper stratified layer, and D is the total depth. Then the 
geopotential gradient rp ∂∂ /  in the axisymmetric flow can be expressed from (5) to include the 
deep flow effects in (1). In this way we consider the deep flow feedback on inertially pulsating 
vortex in the upper layer. 
 
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007 
3 
3 The Structure of Pulson Solutions 
 
         First we consider a flow between horizontal level z = 0 and an isopycnal (i.e., constant 
buoyancy) surface z = h assuming  h << D so that the pulson solution can be described in a self-
similar form following Sutyrin (2006) 
 
),(0 ZRMM =  ,              ),(0 ZRB
z
P
−=∂
∂
 ,               ),(),( Sz
S
rZR =
 ,               (7) 
 
where S =1 +  γ sin(ft+λ) – pulsates with inertial frequency, γ is the amplitude of pulsations, and  
λ is the phase. Then the radial velocity does not depend on the vertical coordinate  
 
        )sin(1
)cos(
220 λγ
λγ
++
+
=== ft
ftrf
S
rSUU
&
,                                                                             (8)        
 
while ),(0 ZRSP Φ= , and )(0 RZSh =  are defined by the spatial vortex structure. The relation 
between 0M  and 0Φ  depends on the pulson amplitude γ 
 
        )1(
4
1 220
4
2
0 γ−=∂
Φ∂
−
f
RRR
M
.                                                                                          (9) 
 
      Such exact solutions of PE describe inertially pulsating combinations of a vortical flow and 
IGW which do not adjust to balanced state. The spatial distribution of ),(01 ZRSP Φ= −  in 
coordinates (R, Z) is the same as for the stationary solution with γ = 0, except its amplitude 
pulsates inversely proportional to S in order to provide the mass conservation described by Eq. 
(3).  Correspondingly, the azimuthal velocity calculated from Eq. (9) for γ  > 0 deviates from 
stationary gradient balance to compensate impact of pulsating radial velocity. Such solution 
describes anticyclonic (warm-core) lens-like vortex with all isopycnals outcropping at the level 
0=z  at variable radial distances. The actual maximum vortex radius at this level pulsates with 
time as )(00 tSRr = , where 0R  is defined by 0)( 00 =RZ .  
      In particular, for a parabolic radial profile of )/1( 2020 RRZZ m −=  and constant 0B , from 
Eq. (7) we obtain )( 000 ZZB −=Φ  so that from Eq. (9) the azimuthal velocity has the form 
 
        
[ ]SAfRSfRMVRS m −−−=−= 22202/1 122 γ                                                        (10) 
 
where 20
2
0 /8 RfBZA mm =  characterizes the nondimensional vortex amplitude and such 
solution exists for any mA−< 1γ . At the moment of maximum contraction γ−=1S , the 
radial velocity becomes zero in (8), and the azimuthal velocity becomes zero for a particular 
value of 2/)211( mA−−=γ . In this case the solution describes the evolution of the lens-like 
perturbation without initial velocity.   
 
 
18ème Congrès Français de Mécanique Grenoble, 27-31 août 2007 
4 
 
4 Deep Flow Feedback on the Upper Layer Vortex 
 
         Using asymptotic expansion in DZm / , in the leading order the deep velocity is obtained 
from (5), (6) and (8)  
 
         
D
Z
S
Sr
u 022
&
−= ,                              ∫−= tudtfv 0 ,                                                        (11)  
 
To calculate the azimuthal velocity in (11), it is convenient to start at the moment of maximum 
contraction setting 2/πλ −=  so that )cos(1 ftS γ−= , then for a parabolic profile we obtain  
 
  )(11
2
111
2 20
2
FSH
SFR
r
SFH
Zfr
v m −⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛
+−⎟⎠
⎞⎜⎝
⎛
−= ,                                              (12) 
 
where )/,1max()( 202 RrrF γ−=  and H is the Heaviside function. Note, only periods when 
expanding lens generates the radial velocity in deep layer are taken into account in this 
expression for γγ +<<− 11 00 RrR . Finally, the deep geopotential gradient is found from 
 
  
t
ufv
r
p
∂
∂
−=
∂
∂
.                                                                                                            (13) 
 
       The correction to the upper layer motion is calculated by linearizing (1) – (4) relative to the 
pulson solution using the deep geopotential gradient (13) as a forcing term.  
 
5 Conclusions 
 
        Here we calculated the flow patterns generated in homogeneous fluid below stratified 
pulsating lens-like vortex described by (11).  The feedback on the upper layer vortex is provided 
by the geopotential gradient according to (13).  
 
References 
 
Reznik, G. M., Zeitlin, V. & Ben Jelloul, M. 2001 Nonlinear theory of geostrophic adjustment. 
     Part 1. Rotating shallow-water model. J. Fluid Mech. 445, 93–120. 
 
Sutyrin, G.G. 2004 Agradient velocity, vortical motion and gravity waves in rotating shallow  
     water model. Quart. J. Roy. Meteorol. Soc., 130, 1977-1989. 
 
Sutyrin, G.G. 2006 A self-similar axisymmetric pulson in rotating stratified fluid. J. Fluid    
     Mech., 560, 243-248. 
 
Zeitlin, V., Reznik, G.M. & Ben Jelloul, M. 2003  Nonlinear theory of geostrophic adjustment. 
     Part 2. Two-layer and continuously stratified primitive equations. J. Fluid Mech. 491, 207–228. 


Generation of inertia-gravity waves by pulsating lens-like axisymmetric vortices

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